Tensor networks (TNs) are a family of computational methods built on graph-structured factorizations of large tensors, which have long represented state-of-the-art methods for the approximate simulation of complex quantum systems on classical computers. The rapid pace of recent advancements in numerical computation, notably the rise of GPU and TPU hardware accelerators, have allowed TN algorithms to scale to even larger quantum simulation problems, and to be employed more broadly for solving machine learning tasks. The ‘quantum-inspired’ nature of TNs permits them to be mapped to parametrized quantum circuits (PQCs), a fact which has inspired recent proposals for enhancing the performance of TN algorithms using near-term quantum devices, as well as enabling joint quantum–classical training frameworks that benefit from the distinct strengths of TN and PQC models. However, the success of any such methods depends on efficient and accurate methods for approximating TN states using realistic quantum circuits, which remains an unresolved question. This work compares a range of novel and previously-developed algorithmic protocols for decomposing matrix product states (MPS) of arbitrary bond dimension into low-depth quantum circuits consisting of stacked linear layers of two-qubit unitaries. These protocols are formed from different combinations of a preexisting analytical decomposition method together with constrained optimization of circuit unitaries, with initialization by the former method helping to avoid poor-quality local minima in the latter optimization process. While all of these protocols have efficient classical runtimes, our experimental results reveal one particular protocol employing sequential growth and optimization of the quantum circuit to outperform all others, with even greater benefits in the setting of limited computational resources. Given these promising results, we expect our proposed decomposition protocol to form a useful ingredient within any joint application of TNs and PQCs, further unlocking the rich and complementary benefits of classical and quantum computation.

中文翻译：

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将矩阵乘积态分解为浅量子电路


张量网络（TN）是一系列基于大张量的图结构分解的计算方法，长期以来一直代表着在经典计算机上近似模拟复杂量子系统的最先进方法。数值计算领域近期的快速发展，特别是 GPU 和 TPU 硬件加速器的兴起，使得 TN 算法能够扩展到更大的量子模拟问题，并更广泛地用于解决机器学习任务。 TN 的“量子启发”性质允许它们映射到参数化量子电路（PQC），这一事实激发了最近关于使用近期量子设备增强 TN 算法性能的提议，以及实现联合量子 -受益于 TN 和 PQC 模型独特优势的经典培训框架。然而，任何此类方法的成功都取决于使用现实量子电路来近似 TN 态的有效且准确的方法，这仍然是一个悬而未决的问题。这项工作比较了一系列新颖的和先前开发的算法协议，用于将任意键维度的矩阵积态（MPS）分解为由两个量子位酉的堆叠线性层组成的低深度量子电路。这些协议是由预先存在的分析分解方法与电路酉的约束优化的不同组合形成的，前一种方法的初始化有助于避免后一种优化过程中质量差的局部最小值。 虽然所有这些协议都具有高效的经典运行时间，但我们的实验结果表明，一种特定的协议采用量子电路的顺序增长和优化，其性能优于所有其他协议，在有限计算资源的情况下具有更大的优势。鉴于这些有希望的结果，我们期望我们提出的分解协议能够在 TN 和 PQC 的任何联合应用中形成有用的成分，进一步释放经典计算和量子计算的丰富且互补的优势。